Fractional Unit Root Tests Allowing for a Structural Change in Trend under Both the Null and Alternative Hypotheses

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1 Fractional Unit Root Tests Allowing for a Structural Change in Trend under Both the Null and Alternative Hypotheses Seong Yeon Chang y Xiamen University Pierre Perron z Boston University November 7, 26 Abstract This paper considers testing procedures for the null hypothesis of a unit root process against the alternative of a fractional process, called a fractional unit root test. We extend the Lagrange Multiplier (LM) tests of Robinson (994) and Tanaa (999), which are locally best invariant and uniformly most powerful. We generalize the LM tests to allow for a slope change in trend with or without a concurrent level shift under both the null and alternative hypotheses. We show that the limit distribution of the proposed LM tests is standard normal. Finite sample simulation experiments show the tests to have good size and power. As an empirical analysis, we apply the tests to the Consumer Price Indices of the G7 countries. JEL Classi cation: C22 Keywords: Hypothesis testing; LM test; Spurious brea; Trend function; Slope Change in Trend. We are grateful to Tomoyoshi Yabu and participants at the th International Symposium on Econometric Theory and Applications (SETA 25) for constructive comments, and Fabrizio Iacone and Dupa Kim for sharing their GAUSS and MATLAB codes. y Wang Yanan Institute for Studies in Economics, Deartment of Statistics, MOE Key Laboratory of Econometrics and Fujian Key Laboratory of Statistical Science, Xiamen University, Xiamen, Fujian 365, China (sychang@xmu.edu.cn) z Department of Economics, Boston University, 27 Bay State Rd., Boston MA 225 (perron@bu.edu)

2 Introduction Non-stationarity in economic time series is a pervasive feature. In order to carry proper inference, it is important to nd the exact features that lead to this non-stationarity. A unit root process is a well nown example of non-stationary processes and testing for a unit root against stationarity has been a topic of substantial interest from both theoretical and empirical perspectives. Perron (989), however, showed that the Dicey and Fuller (979) type unit root test is biased in favor of a non-rejection of the unit root null hypothesis when the process is trend stationary with a structural change in slope. Perron (989, 99) proposed testing procedures in which a structural brea is allowed under both the null and alternative hypotheses. Later, Christiano (992) and Zivot and Andrews (992) criticized the assumption that the date of the structural brea is nown a priori. In succeeding research, Zivot and Andrews (992), Perron (997), and Vogelsang and Perron (998) treated the brea date as unnown and proposed testing procedures for a unit root. In much wor, especially that of Zivot and Andrews (992), it was common to allow for a structural brea only under the alternative hypothesis, not under the null hypothesis of a unit root. This is very restrictive and can lead to misleading results. Recent advances in testing for and estimating a structural brea in a trend function have made possible the development of unit root tests that allow for a change in trend under both the null and alternative hypotheses. Perron and Zhu (25) established the consistency, rate of convergence, and limiting distribution of the parameter estimates when there is a brea in a trend function with or without a concurrent level shift. Perron and Yabu (29) suggested a testing procedure for structural changes in the trend function of a time series without any prior nowledge of whether the noise component is stationary or has an autoregressive unit root. Building on this wor, Kim and Perron (29) proposed unit root testing procedures which allow for a structural change under both the null and alternative hypotheses. See also Carrion-i-Silvestre, Kim, and Perron (29) for an extension to the case with multiple changes. Fractional processes with the order of integration d > :5 are non-stationary as well. Standard unit root tests often reject the null hypothesis when the true process is fractionally integrated with d 2 (:5; ). This can lead to the misleading conclusion that the process of interest is stationary. This motivated researchers to introduce unit root tests which are powerful against the alternative hypothesis of a fractional process. Robinson (99, 994) proposed a Lagrange Multiplier (LM) test of a unit root process against a fractional process, i.e., I() vs. I(d) with d 2 [; ), in the frequency domain. Tanaa (999) suggested a LM test in the time domain and showed that it is locally best invariant and uniformly most powerful. Dolado, Gonzalo, and Mayoral (22) introduced a Wald type unit root test

3 against the alternative of fractional integration. This test is based on the Dicey and Fuller (979) type test using an auxiliary regression with a consistent estimate of the integration order. Lobato and Velasco (27) established a Wald type test which is more e cient and is asymptotically equivalent to the LM test. Recently, Cho, Amsler, and Schmidt (25) suggested combining the test of Kwiatowsi, Phillips, Schmidt, and Shin (992) and a unit root test to test the null of integer integration, i.e., I() or I() against the alternative of fractional integration, i.e., I(d); d 2 (; ). In this line of wor, the process of interest has been limited to either a random wal or a purely fractional process. Lobato and Velasco (27) considered short-run dynamics in the process. Dolado, Gonzalo, and Mayoral (28) extended the wor of Dolado et al. (22) and Lobato and Velasco (27) to incorporate some deterministic components, for instance, a constant and a linear trend function. Our main contribution is to extend the LM test for a fractional unit root to allow for a structural change in a trend function under both the null and alternative hypotheses. This extension has some advantages: (i) it imposes a symmetric treatment of the nature of the deterministic trend under both the null and alternative hypotheses, (ii) it is not required to distinguish long memory from structural change; (iii) the power of fractional unit root tests can be substantially improved when a brea is actually present. We consider linear trend models in which a structural change in slope occurs with or without a concurrent level shift. The rest of this paper is organized as follows. In Section 2, we rst introduce fractional processes and the Lagrange Multiplier test of Tanaa (999) along with preliminary results to be used subsequently. In Section 3, the LM tests are generalized to allow for a structural brea in trend under both the null and alternative hypotheses. Extensions to processes with short-run dynamics are discussed in Section 4. The results of simulation experiments about the size and power of the tests are presented in Section 5. As an empirical application, we test for a fractional unit root in the Consumer Price Indices (CPI) of the G7 countries in Section 6. Concluding remars are in Section 7. All mathematical proofs are relegated to an Appendix. 2 Lagrange Multiplier Test For an integer d = ; 2; : : :, the operator d = ( L) d denotes the di erencing operator with the usual lag operator L, i.e., LX t X t, X t = X t X t, 2 X t = ( 2L + L 2 )X t = X t 2X t + X t 2, and so on. For a non-integer real number d >, the di erence operator d = ( L) d is de ned by means of the binomial expansion X d X d = ( L) = (d)l ; = 2 =

4 where d d!!(d )! d (d) ( ) = d (d ) (d + ) = ; ( ) 2 Y s= s d s ( d) = ( + ) ( d) ; and (d) =. Recall that x! with () the gamma function, so that (d) = d (x + ), x = ; ; : : :, and for = ; 2;, < x <, (x ) is de ned as (x) = (x ) (x ) = = (x ) (x ) (x ). To de ne a fractional process, we use the notation of Robinson (25). Let f t ; t = ; ; : : :g be a short-memory zero-mean covariance stationary process, with spectral density that is bounded and bounded away from zero. For d 2 ( :5; :5), t = d t ; t = ; ; : : : ; () is covariance stationary and invertible. The truncated version of t is de ned as # t = t t ; t = ; ; : : : ; (2) where A is the indicator function for the event A. For an integer m, u t = m # t ; t = ; ; : : : (3) is called a type I I(m + d) process. Let D[; ] be the space of functions on [; ] which are right continuous and have left limits, equipped with the Sorohod topology. Let p! denote convergence in probability and d! convergence in distribution. Denote by [a] the integer part of a 2 R. The order of integration is d = d + m with d 2 ( :5; :5) and m 2 f; g. Remar Marinucci and Robinson (999) de ned type I and type II fractional Brownian motions with d 2 ( :5; :5) on D[; ], respectively as follows: Z B I (t) = [(t s) d ( s) d ]db(s) + (d + ) Z t B II (t) = (t s)db(s) ; (d + ) where B() denotes the standard Brownian motion. Z t (t s)db(s) ; Furthermore, Robinson (25) and Davidson and Hashimzade (29) pointed out that asymptotic results vary depending on the de nition of fractional Brownian motions considered, which requires one to design simulation experiments in accordance with the particular type used. The restriction that d 6= :5 is standard in the long memory literature. Tanaa (999) showed that the case with d = :5 needs to be treated separately from the case with d 6= :5. 3

5 Now we consider a fractional unit root test. Under the null hypothesis, fu t g is a unit root process, that is, H : d =, i.e., d = and m =. The alternative hypothesis can be either one sided (H : d > or H : d < ) or two sided (H : d 6= ). Robinson (994) and Tanaa (999) considered the Lagrange Multiplier test in the frequency and time domain, respectively. It is well nown that the LM test is locally best invariant. Further, Tanaa (999) showed that the LM test is locally uniformly most powerful invariant because it achieves the power envelope of all the invariant tests against local alternatives. The test statistic suggested in Tanaa (999) is LM = p T r 6 XT 2 ; = where = ( P T j=+ u j u j )= P T j= (u j) 2 is the th order autocorrelation of the residuals u t. Local alternatives to the null hypothesis are often considered in the literature, with the integration order de ned as d = + T =2 with xed. We state the limiting distribution of the LM test under local alternatives as it will be relevant for subsequent derivations. Lemma (Theorem 3. in Tanaa, 999) Under the assumption that u t is generated by (3) with d = + T =2 and xed, it holds that, as T!, LM d! N ( p 2 =6; ). 3 Deterministic Components Allowing for a Structural Change In this section, we extend the LM test for a fractional unit root to allow for a structural change in trend with or without a concurrent level shift. We consider the time series of interest y t as consisting of a deterministic component (f t ) and fractionally integrated errors. The data-generating process (DGP) is speci ed as y t = f t + u t : For u t, we impose E(u t ) = and the following assumption. Assumption u t is a type I I(m + d) process which is de ned in ()-(3). Moreover, the short-memory zero-mean covariance stationary process t is assumed to be independent and identically distributed (i:i:d:) with zero mean and nite variance. The i:i:d: assumption on the short-memory process t will be relaxed later to allow for short-run dynamics. The unit root null hypothesis corresponds to the case with m = and d =, which implies that u t is a weighted sum of t. 4

6 3. Change in Mean We rst consider the case where y t experiences a level shift at an unnown time T b. The DGP is speci ed as where C t is a dummy variable for a level shift de ned by: 8 < if t T b ; C t = : if t > T b ; y t = + b C t + u t ; b 6= ; (4) where T b = [T b ] is the true brea date with the corresponding true brea fraction b 2 (; ). Theorem (Change in Mean). generated under the null hypothesis of (4). de ned by: LM M = p T Under Assumption, suppose that the process fy t g is Consider the Lagrange Multiplier test LM M r P 6 T t= ( log y t) y t 2 P T t= (y : t) 2 Under the null hypothesis H : d =, it holds that, as T!, LM M d! N (; ). Theorem implies that Tanaa s (999) LM test is robust to the presence of a level shift. In the following subsection, we consider the LM test in the context of trending series. 3.2 Slope and Intercept Change in Trending Series We now introduce a deterministic time trend in the models. We follow the notation in Kim and Perron (29) (henceforth KP) from which we will use some relevant results. The DGPs are speci ed as. Model A: (Deterministic time trend without a structural change) y t = + t + u t ; (5) 2. Model A: (Level shift) y t = + b C t + t + u t ; (6) 3. Model A2: (Joint broen trend) y t = + t + b B t + u t ; (7) 5

7 where B t is a dummy variable for a slope change in trend given by 8 < if t T b ; B t = : t T b if t > T b ; 4. Model A3: (Locally disjoint broen trend) y t = + b C t + t + b B t + u t : Following KP, we can rewrite Models A-A3 as follows: y t = z(t b ) t + u t = z t; + z(t b ) t;2 2 + u t ; where z t; = (; t), = ( ; ), 8 >< z(t b ) t;2 = >: C t B t ; 2 = (C t ; B t ) 8 >< >: b b for Model A for Model A2 ( b ; b ) for Model A3: In matrix notation, the models de ned previously can be speci ed as Y = Z Tb + U, where Y = [y ; : : : ; y T ], Z Tb = [z(t b ) ; : : : ; z(t b ) T ], = ( ; 2), and U = [u ; : : : ; u T ]. Consider rst Model A where no structural change is allowed. By taing rst di erences, we can rewrite (5) as follows: y t = + u t = + d t t : (8) The ordinary least squares (OLS) estimate of is ^ = T P T t= y t, which is consistent under both H and H. 2 We de ne g y t = y t ^, the OLS residuals from the regression model (8). Theorem 2 (Linear Trend). Under Assumption, suppose that the process fy t g is generated under the null hypothesis of (5). Consider the Lagrange Multiplier test LM T by: LM T = p T r P T 6 t= 2 P T t= log y g gyt t 2 : gyt Under the null hypothesis H : d =, it holds that, as T!, LM T d! N (; ). de ned 2 Under H, ^ is a T =2 -consistent estimator of the slope coe cient. Hosing (996) considered a stationary ARFIMA(p; d; q) process fy t g and showed the wea convergence of the sample mean for d 2 ( :5; :5). It is not di cult to generalize the result to the case where d 2 (:5; ), for which ^ is a T 3=2 d -consistent estimator. 6

8 In what follows, the aim is to devise Lagrange Multiplier tests allowing for a slope change in trend with or without a concurrent level shift. The following assumption is essential to that e ect. Assumption 2 b 6= and b 2 (; ) for some 2 (; =2). Assumption 2 ensures that there is a single slope change in trend and that the pre and post brea samples are not asymptotically negligible, which is a standard assumption needed to derive useful asymptotic results. Model A (level shift only) will be revisited later. The brea date can be estimated by using a global least-squares criterion: ^T b = arg min T 2 Y (I P T )Y; where P T is the matrix that projects on the range space of Z T, i.e., P T = Z T (Z T Z T ) Z T and = [T; ( )T ], < < =2. Note that Z T is the same as Z Tb except replacing T b with a generic brea date T. Perron and Zhu (25) (henceforth, PZ) established the consistency, rate of convergence, and limiting distribution of parameter estimates when the error is an I() process. With Z ^Tb constructed using the estimate ^T b, the OLS estimate of is ^ = (Z ^Tb Z ^Tb ) Z ^Tb Y and the resulting sum of squared residuals is, for an estimated brea fraction ^ s = ^T b =T (the subscript s refers to the fact that we consider a static regression; a dynamic regression with lagged dependent variables will be considered later): S(^ s ) = ^u 2 t = t= t= y t z( ^T b ) t^ 2 = Y (I P ^T b )Y; where P ^Tb is the projection matrix associated with X ^Tb. The rate of convergence of ^ s for Models A2 and A3 is ^ s b = O p (T =2 ) with I() errors (see Theorem 3 in PZ). Chang and Perron (26) derived the consistency and rate of convergence of ^ s when the noise component is a fractional process with the di erencing parameter d 2 ( :5; :5)[(:5; :5). Speci cally, for Models A2 and A3, ^ s b = O p (T =2+d ) if m = and d 2 ( :5; :5). With the consistent estimates (^ s ; ^ ; ^ b ; ^ ; ^ b ), we can construct the detrended process fey t g, and the Lagrange Multiplier test statistic LM T;^s is given by LM T;^s = p r P 6 T t= T ( log ey t) ey t 2 P T t= (ey : t) 2 The convergence rate of the estimate ^ s is not fast enough to guarantee that LM T;^s has the standard normal limit under H. KP faced a similar issue in dealing with unit root tests. 7

9 They introduced a heuristic explanation of the issue involved, which we brie y review. Let ^ = ^T b =T denote an estimate of the brea fraction such that ^ b = O p (T { ) for some {. The detrended series fey t g is given by ey = ^M z Y = ^M z Z(T b ) + ^M z U = ^M z Z(T b ) ^M z U; (9) where Z(T b ) and Z(T b ) 2 are matrices stacing fz(t b ) tg and fz(t b ) t;2g, respectively, and the idempotent matrix ^M z = I ^Pz = I Z( ^T b )[Z( ^T b ) Z( ^T b )] Z( ^T b ). It is obvious that ^M z Z(T b ) 2 2 = only if the true brea date is used in ^M z. In nite samples, ^ 6= b in general, thereby ^M z Z(T b ) 2 2 will not be zero. It turns out that a fast rate of convergence for the estimate of the brea date is needed for the e ect of ^Mz Z(T b ) 2 2 on the Lagrange Multiplier test to become negligible asymptotically. The following proposition provides a d su cient condition under which LM T;^! N (; ) under H. Proposition Suppose that the process fy t g is generated under the null hypothesis of Model d A2 or A3, and that Assumptions -2 hold. Then, it holds that, as T!, LM T;^! N (; ) if ^ b = o p (T =2 ). Proposition implies that the estimate of the brea fraction should converge at a rate faster than T =2. As shown above, ^ s does not satisfy this condition. Hence, we need to consider alternative ways to accelerate the rate of convergence of the estimate of the brea fraction. KP suggested two possible approaches. The rst is based on minimizing the sum of squared residuals (SSR) of a dynamic regression model. This method is similar to that in Hatanaa and Yamada (999). The relevant dynamic regressions are speci ed as follows: y t = y t + D(T b ) t + 2 ttb + z(t b ) t + u t ; (Model A2) y t = y t + D(T b ) t + z(t b ) t + u t ; (Models A and A3) () where D(T b ) t = for t = T b + and otherwise. We obtain, under the null hypothesis, an estimate of the brea fraction ^ d which has a faster rate of convergence such that ^ d b = O p (T ) for Models A2 and A3 (see Proposition in KP). Let LM T;^d denote the Lagrange Multiplier test statistic with ^ d replacing ^ s. It is worth noting that, as discussed by Hatanaa and Yamada (999) and KP, the estimate ^ d has a negative bias in nite samples especially for Model A3. As we shall see, this will a ect the nite sample properties of the tests. The second approach is to use a trimmed data set using a window whose length depends on the sample size and which contains the estimated brea date. The trimmed series then 8

10 consists of the original one with the data points in the window excluded. KP showed that the rate of convergence of ^ s can be increased with the trimmed data set. Suppose that the estimate of the brea fraction satis es ^ b = O p (T a ) for some < a < and the trimming window has length 2w(T ) with w(t ) c T, c > and < a < <. With this speci cation, the length of the window is negligible in the limit compared to the sample size T, but is still large enough to include the true brea date asymptotically. Following KP s suggestion, one proceeds as follows: Estimate the brea fraction ^ s from the original data set and form a window that ranges from T l T (^ s w(t )) to T h T (^ s + w(t )). A trimmed data set is constructed by removing the original data from T l + to T h and then shifting down the data after the window by D(T ) = y Th y Tl. After the trimming and connecting procedures, we now have a new series fy t g, for t = ; : : : ; T ( T 2w(T )T ), de ned by: 8 < y yt t if t T l ; = : y t+th T l D(T ) if t > T l : Test the null hypothesis H : d = using T l as the brea date, i.e., ^ tr = T l =T. The Lagrange Multiplier test statistic is then given by LM T;^tr = p T r P 6 T t= ( log ey t ) ey t 2 P T t= (ey t ) 2 where ey t is the detrended version of yt using the estimate of the brea date T l (or brea fraction ^ tr ). Remar 2 If the window contains either end of the data, then the process fyt g turns out to be Model A (no structural brea), and the statistic in Theorem 2 should be applied to the trimmed data fyt g. The trimmed process fyt g will satisfy the properties of Model A2 regardless of the speci- cation of the original data fy t g, which implies that we can use a common limit distribution. The following proposition states the limiting distribution of the Lagrange Multiplier test based on the trimmed data, which is the same as would be obtained if the brea date was nown in Model A2. 9

11 Proposition 2 Suppose that the process fy t g is generated under the null hypothesis of Model A2 or A3, and that Assumptions -2 hold. Then, it holds that, as T!, LM T;^tr d! N (; ). As shown in KP, under the null hypothesis of a unit root, the estimate of the brea fraction ^ tr = T l =T converges in probability to the true brea fraction at some rate greater than T. Hence, the su cient condition in Proposition is satis ed, so that the proof of Proposition 2 is trivial and omitted. In concluding this section, we consider the case where there is a change in mean, that is, b 6= as in Models A and A3. In Model A, we assume that there is a level shift only, that is, b 6= and b =. Under the null hypothesis, a stochastic trend generated by the I() error process tends to dominate a level shift. Hence, we cannot estimate the brea fraction b consistently because the magnitude of the level shift is asymptotically negligible. In nite samples, we can ignore the level shift if the magnitude of the brea is small. Then Model A can be treated as Model A, and we can follow the testing procedure pertaining to Theorem 2. However, a loss of power is inevitable if a large change in mean is ignored. On the other hand, the level shift can be speci ed as an increasing function of the sample size, i.e., b = c 2 T =2+ for some c 2 > and >. As addressed in Harvey, Leybourne, and Newbold (2), PZ, and KP, this speci cation provides better approximations of the properties of the tests in nite samples when the level shifts are not very small. The models with b = c 2 T =2+ are labeled as Models Ab and A3b, respectively. Proposition 3 Suppose that the process fy t g is generated under the null hypothesis of Model Ab or A3b. Then, LM T;^ diverges as T!. Although the rate of convergence of the estimate of the brea fraction is faster than in the case of a change in slope (see Proposition 7 in KP), Proposition 3 states that the LM tests cannot obtain the standard normal limiting distribution. Hence, the LM test LM T;^, using the critical values from the standard normal distribution, su ers from some liberal size distortions even when j b j is large Using a Pre-test for a Brea in Slope The results of Theorem 2 and Proposition 2 show that the limit distribution of the test is the same whether there is a brea in slope introduced as a regressor or not, even when the DGP speci es that no brea is present. Hence, unlie the case of testing for a unit root 3 Simulation results related to this issue are available upon request.

12 as in KP, theoretically there is no need to carry a pre-test to improve the power of the test. However, Chang and Perron (26) considered Models A2 and A3 with fractionally integrated errors and showed that the so-called spurious brea issue occurs with the order of fractional integration d 2 (; :5) [ (:5; :5). This extended the results on Nunes, Kuan, and Newbold (995) who considered the unit root case. This means that under both the null and alternative hypotheses, if a brea in slope is not present and one is allowed in the regression, the tted model will with large probability suggest the presence of a brea. This could have an e ect on both the size and power of the test. On the one hand, the slope change regressor may induce added liberal size distortions in nite samples because of the over t. On the other hand, since when no brea exists in the DGP it is a super uous regressor, power maybe be reduced. Hence, it may be the case that in nite samples it is bene cial to use a pre-test for a change in slope and try to choose between models (5) and (7). Since a test for a change in mean will be inconsistent, there is no point in trying to distinguish between models (5) and (6) or between model (4) and the corresponding one without the change in mean. Iacone, Leybourne, and Taylor (23) suggested a sup-wald type test (SW) for Model A2. In particular, it is robust to any order of fractional integration d located in an interval [; :5) excluding the boundary case.5. More precisely, given their recommended choice for the bandwidth when constructing the local Whittle estimate of d, their test is consistent for values of d in the interval [; :32], though we believe the proof can be modi ed to allow the interval [; :5). It follows the generalized least squares approach to construct the test statistic for a structural change in trend by taing d -di erences from the data. To mae the test feasible, the fully extended local whittle (FELW) estimator ^d F ELW of Abadir, Distaso, and Giratis (27) is considered. While the FELW estimator is constructed under the null hypothesis of no structural change, Iacone et al. (23) showed that it also satis es the necessary condition for consistency even with a local brea in trend. Since the true brea date is unnown a priori, the nal statistic SW uses the sup functional of Andrews (993) across all admissible brea dates. This test is asymptotically size controlled for all d s in the prescribed range. Using this pre-test, we can then de ne the alternative estimate of the brea fraction e = ^ SW> where is the critical value for the SW test with a nominal size p%. Given that SW is a consistent test, plim e T! = b if ^ is a consistent estimate of b. If there is no brea in the DGP, we can expect that p% of the estimates s e are nonzero. In order to obtain a consistent estimate of b under the null of no structural brea, we assume that the critical value is a function of the sample size T. Since SW = O p (T % ); % > with a local brea, let = ct % for < < %. This speci cation introduced in KP is useful

13 because it does not have any e ect on the consistency of the test SW and does guarantee that plim T! e = when no brea is present. Hence, based on the consistency of e, it is recommended to use LM T if e = and LM T;^ if e 6=. The LM test statistics with the pre-test are denoted by LM p, LM p, and LM p. Whether using a pre-test is T;^ s T;^ d T;^ tr bene cial will be assessed later via simulation experiments about the size and power of the tests. 4 Short-Run Dynamics We now relax Assumption to introduce short-run dynamics in the noise component. A zero-mean short-memory covariance stationary process t can be represented as a one-sided moving average: t = P j= j t j ; t = ; ; : : : ; where =, P 2 j= j <, () P j= j, and f t, t = ; ; : : :g are i.i.d. random variables with mean zero. A special case of interest is an autoregressive moving average (ARMA(p; q)) process given by (L) t = (L) t. In order to implement the Lagrange Multiplier test, we rst estimate the parameters = ( ; : : : ; p ; ; : : : ; q ) consistently. Then under the null hypothesis, we can construct ^ t = ^(L)^(L) ( L) d ^u t where ^u t is the OLS residuals from the model considered, whereas ^(L) and ^(L) are estimated from (L)( L) d ^u t = (L) t, using d =. With short-run dynamics in the noise component, we consider the following test statistic, LM ^ = p XT T ^ ; where ^ is the th order autocorrelation of residuals ^ ; : : : ;^ T. Tanaa (999) derived an important result related to this statistic when no brea is present. Lemma 2 (Tanaa, 999, Theorem 3.3) Under local alternatives, that is, d = + T =2 with xed, it holds that, as T!, LM^! d N (! 2 ;! 2 ), where! 2 = 2 6 = ( ; : : : ; p ; ; : : : ; q ) ( ; : : : ; p ; ; : : : ; q ) ; i = X j= j g j i; i = X j=i j h j i; with g j and h j the coe cients of L j in the expansion of =(L) and =(L), respectively, and the Fisher information matrix for and. Remar 3 Note that! 2 < 2 =6, hence comparing Lemmas and 2, the LM test has lower local asymptotic power in the presence of short-run dynamics of any ind. As will be shown via simulations, the loss in power can be substantial. It remains, nevertheless, inevitable. 2

14 With the maximum lielihood estimate ^!, we show that ^ LM=^! d! N (; ) as T! under the null hypothesis. In particular, when p = or q =, it is easy to compute ^!. Since v j = &v j + j in both cases, g j = & j, = X j= j &j = & log( &); and = & 2 : Hence, we have! 2 = 2 & 2 (log( &)) 2 ; 6 & 2 and ^! can be computed using ^&. All these results remain valid for all trending models with a brea considered. The relevant correction needed is a simple scaling by ^! so that the test becomes LM T;^ LM T;^ =^!. Proposition 4 Suppose that the process fy t g is generated under the null hypothesis of Model A2 or A3, and that Assumptions -2 hold with t being an ARMA(p,q) process. Then, it holds that, as T!, LM d! N (; ) if ^ T;^ b = o p (T =2 ) and ^!! = o p (). The su cient conditions in Proposition 4 follow from Lemma 2 and Proposition, hence the proof is omitted. The nite sample performance of LM T;^ with ^ 2 f^ s ; ^ d ; ^ tr g allowing for a structural brea under both the null and alternative hypotheses will be examined in the next section. 5 Simulation Experiments In this section, we present results from simulation experiments to illustrate the various theoretical results. Throughout the simulations, the true brea fraction is set to b = :5. The DGP is speci ed as y t = f t + u t where 8 + t for Model A; >< b C t for Model A; f t = b B t for Model A2; >: b C t + b B t for Model A3; and u t = # t = t t, t = ; ; 2; : : :, where t is a short-memory zero-mean covariance stationary process that will be speci ed below. We set some parameters as follows: = :72, = :3, b =, and b =. The con gurations are the same as those in PZ, chosen to obtain distributions that easily reveal the main features of interest. In all cases, the results are obtained via, replications. Also, 5% nominal size tests are considered. 3

15 First, to illustrate the e ect of a structural brea on the power of the fractional unit root test, we consider two di erent models when a structural change in slope is allowed in the DGP; (i) Model A (which ignores a relevant slope change) and (ii) Model A2 (which is well speci ed). The results are provided in Table. It is clear that the power of LM T is much lower than that of LM T;^d, which supports the fact that a structural brea in the DGP should be allowed when testing for a fractional unit root. Tables 2-5 present the rejection probabilities of the tests LM T and LM T;^ at the 5% signi cance level when t i:i:d: N (; ). In Table 2, no structural change is allowed in the DGP (Model A), i.e., y t = + t + u t. The size of LM T is well controlled, which is.5 and.6 with the sample sizes T = 5; 5, respectively. Table 3 reports the results for Model A. The brea fraction is not estimated consistently because the level shift is negligible compared to the stochastic trend induced by the I() errors. Hence, LM T;^s and LM T;^d su er from severe size distortion while LM T maintains size close to the nominal level 5%. Table 4 presents the results pertaining to Model A2. We also consider the test based on trimmed data, LM T;^tr. The test LM T;^d is size controlled while the others show minor size distortion. However, the power of LM T;^d is always lower than that of the other two tests. Table 5 presents the results pertaining to Model A3. Here, we set b = to consider the e ect of an irrelevant level shift. Notice that LM T;^s exhibits liberal size distortion and also shows considerable size distortion. As Chang and Perron (26) noted, the LM T;^d estimate of the brea date shows a pattern of bi-modality when an irrelevant level shift is introduced. This phenomenon is referred to the contamination e ect because the irrelevant level shift can mae the estimate of the true brea date less precise. By construction, the contamination e ect is marginal on LM T;^tr, whose exact size is 6.7% when T = 5. In Tables 6-8, we provide simulation results when the errors have short-run dynamics, i.e., ( L) t = t, t i:i:d: N (; ). We set the value of the autoregressive (AR) parameter at 2 f :5; ; :3; :6; :8g. When =, we can compare the loss of power caused by allowing for dynamics when none is present. The other parameters remain unchanged. Table 6 reports the size and power of the Lagrange Multiplier tests pertaining to Model A, LM T. It is well size controlled with less persistent AR parameters 2 f; :3g, but it is very conservative with a higher AR coe cient 2 f:6; :8g while it shows liberal size distortions with = :5. We nd some interesting features in terms of power. First, power is higher when the AR parameter is negative (in part due to the liberal size distortions). Second, as becomes positive and large, power shrins considerably. In particular, the loss of power is substantial when the AR parameter has increased from.6 to.8. This implies that a su ciently large time span is needed to distinguish fractional integration from wea 4

16 dependence. Comparing Table 3 with Table 6, for the = case, it is obvious that power is substantially lower when an irrelevant AR parameter is introduced. This result suggests that selecting the number of lags in the noise component is crucial to obtain good power. Lastly, with a persistent AR parameter = :8, the LM tests have non-monotonic power, that is, power does not increase when the order of integration d moves away from the null of a unit root. As also discussed in Lobato and Velasco (27), it is di cult to distinguish fractional integration from a highly persistent stationary short-memory process. Table 7 reports the results pertaining to Model A2. They show similar patterns as for Model A. It is noticeable that LM T;^ d performs well in terms of size across all cases while its power is always lower than that of the other tests. Table 8 presents the results for Model A3. LM T;^ d has size distortion even with negative and less persistent AR parameters. This happens because when using the dynamic regression to estimate the brea fraction, the estimate ^ d is negatively biased for Model A3. Among the three LM statistics, based on the trimmed data performs well in nite samples. The size of is well controlled across various values of while and LM T;^ d show liberal size distortions. Moreover, the power loss of is minor relative to the other tests. Hence, LM T;^tr and are the recommended tests in practice. 5. The Size and Power When a Pre-test is Used In Figures -4, we present the size and power of the LM tests as the slope change parameter ( b ) changes in Models A2 and A3 with and without the pre-test being used. As a pre-test, we use the SW test of Iacone et al. (23) at the nominal 5% level. We only consider the version of the LM statistics based on the trimmed estimate of the brea fraction, denoted LM T;^tr and LM p when no short-run dynamics is allowed and by T;^ tr and LM p T;^ tr when an AR() structure is allowed. To assess the extent of the di erences in the size distortion and power, we also report the infeasible LM test based on the true value of brea fraction, denoted LM T;b and LM T; b. The results are presented in Figure for Model A2 (no short-run dynamics), Figure 2 for Model A2, and in Figures 3 and 4 for Model A2 and A3 with short-run dynamics of the form ( :3L) t = t, t i:i:d: N (; ). For LM p T;^ tr and LM p, the trimming window contains 6 observations (the simulation results are not T;^ tr sensitive to the length of the window). The magnitude of brea in the slope of the trend b varies from -4 to 4 in increments of.2. We set b = for Model A3. The sample sizes are T = 5, 3 and the number of replications is, for each value of the parameters. One sided tests against the alternative hypothesis H : d < are constructed at the nominal 5% level. For power, d is set to.8. 5

17 The results for Model A2, presented in Figure, show rst that the version without the pre-test exhibits some liberal size distortions when b is near, which reduce when T increase, though remain noticeable even with T = 3. On the other hand, the version with the pre-test exhibits conservative size-distortions when b is near but not equal to, which again reduce but remain noticeable when T = 3. The most drastic di erences occur when considering the power of the tests. The power of the version without the pre-test is slightly below but near to the power of the version with the true brea fraction when T = 5 for all values of b. When T = 3, the power functions are nearly the same. Things are very di erent when the version with the pre-test is used. When b is near but not equal to zero the power reduces drastically creating pronounced power valleys. This reduction in power alleviates somewhat when T = 3 but remains important. This is due to the fact that for low values of b the SW test of Iacone et al. (23) is not powerful enough so that a change in slope regressor is not included. Yet, the magnitude of the change in slope is big enough to induce a considerable loss in power. This is ain to the problem faced by the Kim and Perron (29) test in the context of testing for a unit root. The results for Model A3, presented in Figure 2, show a similar picture. This is also the case when considering the tests and LM p with serial correlation in the DGP T;^ tr (Figures 3 and 4), though here the size distortions of both tests are somewhat higher, liberal for and conservative for LM p. The power losses of LM p is severe, especially T;^ tr T;^ tr when T = 5. Based on the simulation results, it is recommended to use the LM T;^tr or tests without the pre-test. In our view, the reduction in power when using a pre-test considerably outweigh the di erences in size distortions. The SW test of Iacone et al. (23) is nevertheless still useful to assess the presence of large breas. 6 An Empirical Application We analyze the aggregate price indices of the G7 countries. Monthly seasonally adjusted CPI series were obtained from the OECD Main Economic Indicators. All series are analyzed with a logarithm transformation and are plotted in Figure 5 where the vertical line is the brea date estimated by minimizing the sum of squared residuals from Model A2. The results are presented in Table 9. We only consider Model A2 and use the test for a slope change of Iacone et al. (23). Based on the simulation results, we recommend using the LM tests with ^ tr. We present results with and without short-run dynamics. When dynamics is allowed, an AR() speci cation is used. The time span is from January 969 to December 27 (T = 468). First, the augmented Dicey-Fuller type test (ADF) cannot 6

18 reject the null of a unit root against the alternative of trend stationarity for all G7 countries. On the other hand, the SW test of Iacone et al. (23) detects a change in the slope of the trend. Allowing for a structural change in trend, the fractional unit root tests LM T;^tr and lead to a rejection of the unit root in favor of fractional integration. Speci cally, the test results state that the order of fractional integration is greater than unity for all G7 countries. This result is compatible with that in Gil-Alana (28) where he estimated the order of fractional integration for the U.S. CPI and found that the con dence intervals were located above unity. Hassler and Wolters (995) considered the in ation rates for various countries and found that the order of fractional integration was located in an interval (; :5), that is, the in ation rate is a long-memory process. 7 Conclusion We established testing procedures for a fractional unit root allowing for a structural change under both the null and alternative hypotheses. Following Robinson s (994), Tanaa (999) derived a Lagrange Multiplier test in the time domain, and Dolado et al. (22, 28) and Lobato and Velasco (27) considered Wald-type tests for a unit root null hypothesis against fractional integration. Although Dolado et al. (28) introduced deterministic components, the case with a structural brea in trend has not been considered in the literature. In contrast to the large amount of wor related to testing the null hypothesis of long-memory against the alternative of stationarity with level shifts, and vice versa, wor related to a fractional unit root test allowing for a structural brea in trend is more scarce. To the best of our nowledge, this paper is the rst that addresses testing for a fractional process allowing a structural brea under both the null and alternative hypotheses. Fractional unit root tests allowing for a structural brea under both the null and alternative hypotheses have some desirable features: (i) given that economic variables are often subject to structural changes, our approach imposes a symmetric treatment of the change under both the null and alternative hypotheses; (ii) it is not required to distinguish long memory from structural change; (iii) the power of fractional unit root tests can be substantially improved when a brea is actually present. Under some conditions, the proposed LM test statistics have the standard normal limit under the null hypothesis. Simulation experiments con rmed that the tests have good size and power. Hence, we believe that our procedures o er useful complements to existing tests and should be used in practical applications. 7

19 Appendix Proof of Theorem. The DGP is speci ed by (4), that is, for t = ; : : : ; T, y t = + b C t + u t ; b 6= : Under H, tae rst di erences of y t and de ne D(T b ) t = if t = T b + and otherwise. Then, and T T (y t ) 2 = T t= ( b D(T b ) t + t ) 2 t= = T 2 t + 2 t= b D(T b ) t t + t=! = T 2 t + 2 b Tb b t= = T 2 t + o p (); t= ( log y j ) y j = T j=2 j=2 = T j=2 = T j=2 + T j=2 j X = j X = j X =! 2 bd(t b ) 2 t t=! ( bd(t b ) j + j ) ( b D(T b ) j + j ) j X bd(t b ) j + 2 bd(t b ) j j X = j A + A 2 + A 3 + A 4 :!! = j! D(T b ) j + T b D(T b ) j + T ( b D(T b ) j + j ) j=2 j=2 j X = j X = j bd(t b ) j! j! j It is easy to show that A =. For A 2, we have 2 T (T b +) (A 2 ) 2 = 2 b T 2 A T ++ 2 b T 2 = = 2 2 T (T b +) b 2 A 6 T 2 T ++ = 2 b 6 T = T (T b +) 2 T (T b +) X = 2 T ++ X T (T b +) 2 T ++ = A A T = O p(t ) = o p (); 8

20 where the rst inequality follows from the Cauchy-Schwarz inequality, and the second inequality holds because P = 2 = 2 =6 (see Tanaa, 999). Because t is a short-memory zero-mean covariance stationary process, it is straightforward to show that T P T (T b +) = 2 T ++ = O p (). By the continuous mapping theorem, ja 2 j = o p (), which implies that A 2 = o p (). Similarly, Hence, (A 3 ) 2 = 2 b T b 6 XT b T LM M = p r PT 6 T T 2 T b + = XT b 2 T b + = j=2! 2 2! Pj = j b T 2 T = o p(): T P T j=2 2 j + o p() XT b = j + o p ()! 2 = p T XT b 2 T b + = r 6 2 The result in Lemma and Slutsy s theorem imply that LM d! N (; ). =! + o p (): Proof of Theorem 2. Under H : d =, y t = + u t = + t t. De ne gy t = y t ^. First, consider the denominator of LM T. Under H, conditioning on y =, T 2 gyt = T t= t= y t ^ 2 = T 2 ^ + t t= = ( ^ ) 2 + 2( ^ )T t + T t= = O p (T ) + o p () + T 2 t = T where we use the fact that ^ = O p (T =2 ), and T P T t= t large numbers. Second, the numerator of LM T is given by! T log y g j X gyj j = T gy j j=2 j=2 = T j=2 = T j=2 = j X = j X = t= ^y j t= 2 t 2 t + o p (); t= p! by the wea law of y! j ^ y j ^! ^ + j ^ + j 9

21 2 = ^ T j X j=2 = + ^ T j=2 = B + B 2 + B 3 + T where we use the expansion i = ; 2; 3. + j X j=2 = j j X =! j ^ T j=2 + T! j ; j=2 j X = j X =! j! j log = L + 2 L2 + 3 L3 +. We show that B i = o p () for 2 B ^ T T (log T + + T ) = O p (T ) (log T + + T ) = o p (); B 2 = ^ j A = O p (T =2 )o p () = o p (); = j=+ B 3 = ^ = j= j A = O p (T =2 )o p () = o p (); where is the Euler-Mascheroni constant and T =(2T ) which approaches as T!. The results for B 2 and B 3 follow from the arguments used in the proof of Theorem for the terms A 2 and A 3. Hence, Then, under H, T j=2 This completes the proof. log g y j gyj = T LM T = = Proof of Propositions and 3. j=2 T P T Pj j=2 = j j X = j T P T t= 2 t + o p () T P T Pj j=2 = j j T P T t= 2 t! j + o p () + o p (): j j + o p (): When the estimate of the brea date is consistent, the proof is trivial and omitted. We here focus on having a consistent estimate of the brea fraction at some rate T { for < { : Speci cally, suppose that the estimate of the brea fraction ^ satis es that ^ b = o p (T { ) for < {. For all models A-A3, we have a detrended sequence fey t g based on the OLS method in PZ. The Lagrange Multiplier test 2

22 statistic is given by: LM T;^ = p r P 6 T t=2 T ( log ey t) ey t 2 P T t= (ey t) 2 = p T r P T 6 t=2 2 P t = ey t eyt P T t= (ey : t) 2 We show that the stochastic orders of terms associated with the deterministic time trend are smaller than those of terms associated with the error process. We can write (9) as: ey = ^M z Z(T b ) + ^M z U = ^M z Z(T b ) ^M z U = f M + e U; and Y e = M f + U. e Since LM T;^ is a functional of Y e and subvectors of Y e, it su ces to consider the inner product of Y e Y e, that is, Y e Y e = M f M f + 2M f U e + U e U: e Note that we only need to chec the stochastic order of Y e Y e because the lag of order is controlled to be small relative to the sample size T. We want to show that the term U e U e dominates the others. It is straightforward to show that U e U e = O p (T ) uniformly over all admissible brea dates T b 2 ft; ( )T g for some 2 (; =2) in all models. When M f M f has a smaller order of magnitude compared to that of U e U, e so does M f U e by the Cauchy-Schwartz inequality. Further, the order of magnitude of M f M f cannot be greater than that of M f M. f The order of magnitude of M f M f is O p (T 2 2{ ) for Models A2 and A3, which implies that the brea fraction should be estimated consistently at some rate greater than T =2. On the other hand, for Model Ab, the stochastic order of M f M f is O p (T +2 ). Hence, the orders of magnitude of terms associated with the deterministic trend are greater than those of the terms associated with the error process, thereby the LM statistic diverges as T!. 2

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